Bifurcation Of Traveling Wave Solutions For Two Classes Of Nonlinear Schr?dinger Equations

The research on bifurcation of traveling wave solutions for nonlinear differential equation and dynamical system is the international frontier subject and hot issue in the field of nonlinear dynamics.Nonlinear Schr?dinger equation is a class of nonlinear model which modelled combined with differential equation and the concept of material waves.It is widely used in many fields such as nonlinear optical fiber,ionic sound waves of plasma,optical metamaterials and optical fibre communication.By using traveling wave transformation,the nonlinear Schr?dinger equation is transformed into nonlinear traveling wave system.In particular,“solitary wave” phenomenon in smooth traveling wave system and “new wave” phenomenon caused by special properties of singular nonlinear traveling wave system with singular straight curves are closely related to the complex phenomena in many real models.Bifurcation of traveling wave solutions and dynamical behavior for the quintic derivative nonlinear Schr?dinger equation and the nonlinear Schr?dinger equation in optical metamaterials are studied in this paper,the research contents of this dissertation are as follows:(1)The forms of wave functions are proposed based on the characteristics of the two equations.By using the traveling wave transformation and separating the real and imaginary parts of the results,two equations are transformed into their corresponding nonlinear traveling wave systems by theories and methods of dynamical system such as invariant manifold.The canonical system of singular traveling wave system is obtained by time scale transformation.(2)Based on the theories of bifurcation for planar dynamical system,the equilibrium bifurcation,dynamical behavior and phase portrait bifurcation for smooth traveling wave system,singular traveling wave system and its canonical system are studied by the methods which combined with first integral and qualitative analysis.(3)According to the relationship between the phase portraits of traveling wave system and nonlinear waves,the exact traveling wave solutions are calculated by the mathematical tools such as Jacobi elliptic function and hyperbolic function.Then Maple symbol calculation software is used to simulate the profiles of waves,the relationship between wave function and time variable is obtained.